Is the following statement true? Why?

Question

$\frac{a}{b}\neq0 \Rightarrow (a\neq0\land b \neq0)$

At first sight that seems quite obviously true, however, wouldn't $b = 0$ also fit the condition?

$\frac{a}{0}\neq0$

Answer 1

Asumming they are real numbers, because the object $\displaystyle\frac{a}{b}$ does not exists if $b=0$ (because if $b=0$ then $bb^{-1}=b^{-1}b=0\neq 1 \forall b^{-1} \in \mathbb{R}$

Answer 2

The symbol $\frac{a}{0}$ carries no meaning, and thus must be disallowed.

Answer 3

For a formula (a sequence of symbols) to have a truth value, it must be a sentence, i.e., it must make sense.

The formula $\displaystyle \frac{a}{0}\neq 0$ isn't a statement because it doesn't make sense. You can't tell wether it is true or false. It is not true, it is not false, for it is not.

Answer 4

When we write $\frac{a}{b}$, we always mean that $a$ is anything and $b \ne 0$ (since if $b $ is zero, $\frac{a}b$ would be undefined). That is why we still include $b \ne 0 $ besides $a \ne 0$ to remove all the possible ambiguity.